On the purity of the branch locus

C ^OMPOSITIO M ^ATHEMATICA

A ^LLEN A ^LTMAN

S ^TEVEN L. K ^LEIMAN

On the purity of the branch locus

Compositio Mathematica, tome 23, n

^o

4 (1971), p. 461-465

<http://www.numdam.org/item?id=CM_1971__23_4_461_0>

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461

ON THE PURITY OF THE BRANCH LOCUS by

Allen Altman and Steven L. Kleiman

Wolters-Noordhoff Publishing Printed in the Netherlands

Let f : X -

^{Y be a}

quasi-finite morphism

of schemes and U the _open subset of X where

f

^is étale. The various theorems about the

purity

^of

the branch locus

give

conditions for U to be

all

^{of X. We offer a}

simple elementary proof that

^{U = X i n the} rather useful case when Y is smooth

over a

locally

noetherian scheme S and U contains every

point

^of

depth

~

^{1 and} is dense in the fibers over S. The

proof

^is

inspired by

^Zariski’s

original

^method

[5]

^for characteristic 0. After the usual sort of reduc-

tions,

^Y becomes the

spectrum

^{of the}

ring

of formal power series in a vector T of variables. Zariski

proved

^that the functions on X also form

a power series

ring by expanding

^{them in}

Taylor

^{series in T. The} ap-

propriate

differential

operators (1/i!)(~ig/~Ti)

^{first lift}

canonically

^over

U and then extend over all of X because of the

depth

condition.

However, lifting

^these

operators

^{amounts to}

constructing

formal descent data for X

(cf. [2]

^and

[4])

^{and we}

^present

^the

proof

^{from this}

point

^{of view}

without

mentioning

differential

operators, characteristics,

^{or descent}

(and

without

using deeper

^{results from formal}

geometry 1).

The various standard results we need have been collected in

[1 ]

^{and all our references}

below are to this source.

THEOREM

(Purity

^{of the branch}

^locus;

^cf.

^VI, 6.8).

Let S be a

locally

noetherian

scheme,

g : Y - S a smooth

morphism and f :

^{X ~ Y a}

quasi- finite morphism.

^{Let U} be the open subset

of

^X

where f is

^{etale. Assume U}

contains every

point

^{x where}

depth(Ox) ~

^{1 and that}

(U

ⁿ

X(s))

^{is dense}

in the

fiber X(s) for

^{all s in S. Then U = X.}

NOTE.

(i)

^If

g(Y) = S and f(U)

^{contains every}

^{point y}

^{cf Y where}

dim(Oy) ~ dim(S),

^then

automatically (U

ⁿ

X(s))

is dense in

X(s)

^for

all s in S.

(ii)

^If

(g of)(U) = S,

^the

^following

conditions are

equivalent:

(a)

^{X satisfies}

S2 (resp.

^{X is}

normal)

^{and U contains} every

point

^x

where

dim(Ox) ~

^1.

1 However, using ^such results, Grothendieck [3 ] has ^also proved ^{that U = X when}

Y is locally ^a complete intersection and U contains _every point ^of dimension ~ 2.

462

(b)

^{U satisfies}

^{s2 (resp.}

^{U is}

normal)

^{and U contains every}

point x

where

depth(Ox) ~

^1.

(c)

s satisfies

S2 (resp. S

^is

normal)

and U contains _every

point x

where

depth(Ox) ~

^1.

Indeed,

^the

equivalence

^of

(a)

^and

(b)

^results

directly

from the defini- tions

(resp.

and Serre’s

criterion).

^The

equivalence

^of

(b)

^and

(c)

^holds

by (VII, 4.9)

because U ~ S is smooth and

surjective.

(iii)

In view of

(i)

^and

(ii),

the theorem

(applied

^{to X minus the}

components

of codimension one of the branch

locus) implies

^{that if X}

satisfies

S2 (e.g.,

^X

normal)

^and

dim(S) ~ ^1,

then the branch locus of

f has

pure codimension ^1.

PROOF.

By

way of

contradiction,

^assume

U ~

^{X. Let x} be a

generic point

^{of an} irreducible

component

^of

(X- U).

^{We shall}

prove f is

^étale

at x.

Let y = f(x)

^{and s =}

g(y).

^{Consider the flat base}

change Spec(k) -

^S

where k ⁼

Oy .

^The

hypotheses clearly

^{hold for}

f Q

^k

and g

^Q

k; by (VII, 5.11), U (8) k

^is the open ^{set on}

which f Q9 k

^is

etale;

^the

depth

condition holds

by

^{virtue of}

(VII, ^4.2);

^and

^clearly

^U

^{Q k}

is dense in the fibers over

Spec(k).

^{Thus we may assume that S is the}

spectrum

^of

a local

ring k

and that there exists a section h : S ~ Y such that

h(s)

^{= y.}

Note that

0 /Oy

^{is étale if}

(and ^only if) 6xl6y ^is,

^{that Y is an étale}

extension of a

polynomial ring k[T1, ···, Tn] with y lying

^over

(T),

that

depth(Ôx)

⁼

depth(Ox) by (VII, ^4.2)

^{and that}

Ôx

^{is a} localization of

Ox~Oy Ôy. Replace X by Spec( 6 x),

^y

by Spec(Ôy)

^{and k}

^{by k. While g}

^is

no

longer

^{of finite}

type,

^now

Oy ~ k[[T1, ···, Tn]], f is

finite and U ⁼

X-{x}. Furthermore, clearly

^{U contains} every

point

^of

depth ~

^{1 and}

(g o f)(U)

^{= s. Let V =}

(Y-{y}).

^Then

f

^{is étale over and since}

depthOy(B) = depthB(B )

^{where B}

= Ox by (III, ^3.16),

^the open ^{set V} contains _every

point

^{z E Y such that}

depthOz(Bz) ~

^1.

Finally,

^{it suffices} to construct an

isomorphism ^Xo

x s Y X where

X0 =

^{X x}

y s.

^For

then, by (VII, 5.11), Xo/S

^{is étale because U}

= Xo

^x s V is étale over and V ~ S is

surjective

^and

flat;

^whence

X/Y

^{is etale} because Y - S is

surjective

^{and flat. Thus it suffices to} prove the follow-

ing

^theorem

(whose proof

^{will be}

presented

^{after several}

preliminary lemmas).

THEOREM. Let k be a noetherian

ring

^{and A =}

k[[T1, ···, Tn]] a formal

power series

ring.

^{Let B be a}

finite A-algebra

^{which is étale over} every

prime

p

of

^{A where}

depth(Bp) ~

^{1. Then there exists a}

(canonical)

^iso-

morphism

^A

0k B0

^{B where}

Bo =

^k

QA

^B.

DEFINITION. Let k be a

ring, R

^a

k-algebra.

^{The module}

of mth principal

parts

of

^{R over}

^k,

^denoted

P’(R),

^{is defined as}

(R Q9k R)/Im+1

^{where I}

is the

diagonal

^{ideal. It is}

naturally

^filtered

by

the powers of 1.

LEMMA 1. Let k be a

ring,

^{R a} noetherian

k-algebra

^{and S an étale}

extension

of

^R.

(i)

The natural

(R Q9k R)-algebra homomorphism

Vm :

P’(R) ^(8)R

^{S ~}

Pm(S) ^{sending (al}

^Q9

^{a2 )}

^{(D s to al O sa2}

(resp.

^{to sal Q9}

a2 )

^{is an}

isomorphism (where Pm(R)

^is

^regarded

^{as an R-module}

from

^the

right (resp. left)).

(ii)

^{The induced map}

gri(Pm(R)) ^(8)R

^{S ~}

gr’(Pm(S»

^{is an}

isomorphism.

PROOF. In

(i),

^{both filtered modules are}

separated

^and

complete;

^{so it}

suffices to show that the

gri(vm)

^are

isomorphisms.

^Since

SIR

^is

flat,

gri(Pm(R) ^QR S)

^is

isomorphic

^to

gri(Pm(R)) ^OR

^{S. Thus}

(i)

^follows

from

(ii).

Let I

(resp. J)

^{be the}

diagonal

^{ideal of}

(R/k) (resp. (S/k)),

^{and set}

K ⁼

ker(S Qk

^{S - S}

OR S).

^{As in}

(VI,

^{4.9 and}

4.10),

^{X ~ I}

~(R~03BAR) (S ^{(8)k S)}

^since

S/R

^{is flat. Hence}

(Ki/ki+1) ~ (Ii/Ii+1) Q9(RQ9kR) (S ^Q9k S).

Also, (Ki/Ki+1) Q9(SQ9kS)

^{S gé}

(Ji/Ji+1)

^sil1ce

^SIR

^is unramified. There-

fore, (IiiIi + 1) _Q9(RQ9kR)

^{S ~}

(Ji/Ji+ ^1).

^{Since the}

(R ^Qk R)-module

structure of

(Ii/Ii+1)

coincides with the left

(resp. right)

^R-module

structure of

(Ii/Ii+1),

^this

isomorphism

coincides with the induced map.

LEMMA 2. Let R be a noetherian local

ring; P,

^{N two}

finite

^R-modules.

If depth(N) ~ ^2,

^then

depth (HornR(P’ N)) ~

^2.

PROOF. An

N-regular

sequence

(x1, x2)

^is

^easily

^{seen to be}

HomR(P, N)-regular.

LEMMA 3

(cf. ^{VII, 2.10).}

^{Let R be a} noetherian

ring,

^M

a finite

^R-module

and V an open subset

of Spec(R).

(i) ^Suppose

^{V contains every}

point

p where

depth(Mp) = ^0; (e.g.,

^V

contains _every

generic point of Supp(M)

^{and M}

^{satisfies Sl).}

^{Then the}

restriction M -

F(V, M )

^is

injective.

(ii) ^Suppose

^V contains every

point

p where

depth(Mp) ~ ^{1 ; (e.g.,}

^V

contains every

point of codimension ~

1 in

Supp(M)

^{and M}

satisfies S2).

Then M ~

F(V, M)

^is

bijective.

PROOF. To prove

(i),

^{let x ~ M go to zero}

in 0393(V, M).

^{Assume x :0 0.}

Then there exists a

prime p

ⁱⁿ

Ass(Ax). Then pAp

^E

Ass(Apx)

^c

Ass(Mp),

so

depth(Mp) =

0. Hence p E

Tl,

^so

Apx

⁼

^0;

this contradicts

pAp

^E

Ass(Ap x).

To prove

(ii),

^let

f E 0393(V, M).

^{Let E be the ideal of elements s E A}

such that

sf

^{extends to an element x of M. For every}

prime

^{p in V the}

image of f in Mp

^{is a fraction}

^xls,

^{and it follows that}

^{E t-}

^p.

^By (III, 1.5),

464

there exists therefore an element s of E not in any

prime p

^where

depth(Mp)

^{= 0. Let x be an} element of M

extending sf.

Since s is

M-regular, V

^contains every

prime p

^where

depth((M/sM)p)

= 0. Since the

image

^{of x in}

(M/sM)

^{is zero}

^{on V,}

^{it is zero}

^by (i).

^Thus

there exists _{a g} in M such that x ⁼ _sg. Then

s(g -ff)

^{is zero over V.}

Since s is

M-regular, g

⁼

f on V,

^{and the}

proof

^is

complete.

PROOF oF THEOREM. Let P

= lim(Pm(A)).

^{It will suffice} to construct

a

P-isomorphism

^{u : P}

~A

^{B - B}

QA

^{P where in P}

QA B (resp.

^B

^OA P),

P is

regarded

^{as an} A-module via the second

(resp. first)

^factor.

Namely,

define w : A

~k

^{A ~ A}

by w(al ^Q a2) = a2(o)al

^where

a2(0)

^{denotes the}

constant term of _a2. Then

w(I)

is contained in m ⁼

T1A+ ···

⁺

TnA,

so w defines an

A-homomorphism w :

^{P ~} A. Since the

diagram

is

commutative,

^A

O p ( P Q9A B )

^{= A}

~k ( k ~A B). ^Hence, (A Q9p u) : (A Ok Bo) B

^{is the}

required isomorphism.

Since A

= k[[T1, ···, Tn]],

^the

(A ~k A)-module Pm(A), ^regarded

^as

an A-module on the left

(resp. right)

^is

isomorphic

^to

A~r

for some r.

Therefore

(B QA Pm(A)) ~ ^{B~r. Thus,}

^the open ^{set V} of

Spec(A)

^over

which B is

étale,

^contains

all p

^where

depth((B ^QA Pm(A))p) ~

^1.

Regarding

^{the two}

(A Q9kA)-modules pm(A) ^Q9AB

^{and B}

QAPm(A)

^as

A-modules on the

left,

consider M ⁼

HomA(Pm(A) OA B, BQ9Apm(A».

By

^lemma

1,

^{M has a natural section over V.}

By

^lemma

2, V

^con- tains _every

point p

^where

depth(Mp) ~

^{1. So}

^by

^lemma

^3,

^{this section}

extends to an

A-homomorphism

um :

Pm(A) ^Q9A

^{B ~ B}

^QA Pm(A).

^In

fact,

um is an

(A Qk A)-homomorphism

^{since it is on V and we may}

apply 3(i). Similarly,

^{we obtain an}

(A ^{Q k} A)-homomorphism B ^{Q9 A} Pm(A )

-

Pm(A) ^{QA B}

^{which is an inverse to um on}

^{V; hence,}

^{it is a}

^global

inverse. The

isomorphisms

^um

clearly

^{form a}

compatible ^system

^of

maps,

inducing

^the

required P-isomorphism

^{u : P}

~A B ~ B OA

^P.

REFERENCES A. ALTMAN AND S. KLEIMAN

[1] Introduction ^to Grothendieck duality theory, Lecture Notes in Math. Vol. 146, Springer-Verlag, Berlin-Heidelberg-New ^York (1970).

A. GROTHENDIECK

[2 ] Appendix ^to Crystals ^{and the De Rham Coh. of} Sch., ^{in Dix} exposés ^{sur la cohomo-} logie ^des schémas, North-Holland Publ. Co., ^Amsterdam (1968).

A. GROTHENDIECK

[3] "Sem ^{de Geom.} Alg. 2", North-Holland Publ. Co., ^Amsterdam (1968). Exposé ^X Theorem (3.4).

N. KATZ

[4] Nilpotent connections and the Monodromy ^Theorem Applications ^{of a Result of} Turrittin", in Sem. on Degeneration ^of Alg. Var., Inst. for Adv. Study, Princeton,

New Jersey (1970). Remark 8.9.16.

O. ZARISKI

[5] "On ^the purity of the branch locus of algebraic functions," Proc. Nat. Acad. Sci.

U.S.A. 44 (1958),

791-796.

(Oblatum 00-00-000)

Allen Altman Steven L. Kleiman

Department of Mathematica Department of Mathematics

University ^of California, ^San Diego Massachusetts Institute of Technology

La Jolla, Calif. 92037 U.S.A. Cambridge, ^{Mass. 02139 U.S.A.}